scholarly journals The Kirchhoff Index of Toroidal Meshes and Variant Networks

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Jia-Bao Liu ◽  
Xiang-Feng Pan ◽  
Jinde Cao ◽  
Xia Huang
Keyword(s):  
Asymptotic Behavior ◽  
Graph Theory ◽  
Distance Function ◽  
Network Theory ◽  
Spectral Graph Theory ◽  
Electrical Network ◽  
Kirchhoff Index ◽  
Resistance Distance ◽  
Spectral Graph ◽  
Explicit Formulae

The resistance distance is a novel distance function on electrical network theory proposed by Klein and Randić. The Kirchhoff index Kf(G) is the sum of resistance distances between all pairs of vertices inG. In this paper, we established the relationships between the toroidal meshes networkTm×nand its variant networks in terms of the Kirchhoff index via spectral graph theory. Moreover, the explicit formulae for the Kirchhoff indexes ofL(Tm×n),S(Tm×n),T(Tm×n), andC(Tm×n)were proposed, respectively. Finally, the asymptotic behavior of Kirchhoff indexes in those networks is obtained by utilizing the applications of analysis approach.

scholarly journals The Kirchhoff Index of Hypercubes and Related Complex Networks

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Jiabao Liu ◽  
Jinde Cao ◽  
Xiang-Feng Pan ◽  
Ahmed Elaiw
Keyword(s):  
Graph Theory ◽  
Complex Networks ◽  
Laplacian Matrix ◽  
Exact Formula ◽  
Spectral Graph Theory ◽  
Kirchhoff Index ◽  
Resistance Distance ◽  
Spectral Graph ◽  
Relationship Of ◽  
The Relationship

The resistance distance between any two vertices ofGis defined as the network effective resistance between them if each edge ofGis replaced by a unit resistor. The Kirchhoff index Kf(G) is the sum of resistance distances between all the pairs of vertices inG. We firstly provided an exact formula for the Kirchhoff index of the hypercubes networksQnby utilizing spectral graph theory. Moreover, we obtained the relationship of Kirchhoff index between hypercubes networksQnand its three variant networksl(Qn),s(Qn),t(Qn)by deducing the characteristic polynomial of the Laplacian matrix related networks. Finally, the special formulae for the Kirchhoff indexes ofl(Qn),s(Qn), andt(Qn)were proposed, respectively.

scholarly journals The Kirchhoff Index of Folded Hypercubes and Some Variant Networks

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Jiabao Liu ◽  
Xiang-Feng Pan ◽  
Yi Wang ◽  
Jinde Cao
Keyword(s):  
Graph Theory ◽  
Characteristic Polynomial ◽  
Laplacian Matrix ◽  
Spectral Graph Theory ◽  
Kirchhoff Index ◽  
Spectral Graph ◽  
Explicit Formulae ◽  
Folded Hypercube

Then-dimensional folded hypercubeFQnis an important and attractive variant of then-dimensional hypercubeQn, which is obtained fromQnby adding an edge between any pair of vertices complementary edges.FQnis superior toQnin many measurements, such as the diameter ofFQnwhich is⌈n/2⌉, about a half of the diameter in terms ofQn. The Kirchhoff indexKf(G)is the sum of resistance distances between all pairs of vertices inG. In this paper, we established the relationships between the folded hypercubes networksFQnand its three variant networksl(FQn),s(FQn), andt(FQn)on their Kirchhoff index, by deducing the characteristic polynomial of the Laplacian matrix in spectral graph theory. Moreover, the explicit formulae for the Kirchhoff indexes ofFQn,l(FQn),s(FQn), andt(FQn)were proposed, respectively.

Resistance Distances and Kirchhoff Index in Generalised Join Graphs

2017 ◽  
Vol 72 (3) ◽  
pp. 207-215 ◽  
Author(s):  
Haiyan Chen
Keyword(s):  
Complete Graph ◽  
Connected Graph ◽  
Effective Resistance ◽  
Electrical Network ◽  
Kirchhoff Index ◽  
Resistance Distance ◽  
Join Graph ◽  
Explicit Formulae

AbstractThe resistance distance between any two vertices of a connected graph is defined as the effective resistance between them in the electrical network constructed from the graph by replacing each edge with a unit resistor. The Kirchhoff index of a graph is defined as the sum of all the resistance distances between any pair of vertices of the graph. Let G=H[G1, G2, …, Gk ] be the generalised join graph of G1, G2, …, Gk determined by H. In this paper, we first give formulae for resistance distances and Kirchhoff index of G in terms of parameters of ${G'_i}s$ and H. Then, we show that computing resistance distances and Kirchhoff index of G can be decomposed into simpler ones. Finally, we obtain explicit formulae for resistance distances and Kirchhoff index of G when ${G'_i}s$ and H take some special graphs, such as the complete graph, the path, and the cycle.

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